Common tiling complements in large-prime extensions of finite cyclic groups

Let R ≥ 2 and let q > R be prime. We prove that the common tiling-complement property for all families of sets with a fixed spectrum is equivalent in the cyclic groups of orders R and qeR for every e ≥ 1. The proof expands the shared spectrum and all its sources by complementary digits. A rank estimate then shows that every terminal frequency fiber is a spectrum of every terminal source fiber. A separate stacking argument relates the common-complement property at order R to spectral tiling at order qR. These results can be iterated along a chain of successively large prime powers.